HRABOVE, Ukraine—Outrage grew Saturday over the rebel hold on the site of the Malaysia Airlines Flight 17 crash that killed 298 people in eastern Ukraine, with European leaders calling for Russian President Vladimir Putin to do more to use his influence to get them to hand over the site.

World leaders and the rebels themselves have called for a transparent, credible investigation into the downing. But two days after the disaster, there has been little progress, with the site disorderly and investigations hampered by its...